Computing the expectation from given probabilities.

Question

I am given the following. Let $(X_n)$ be a sequence of random variables on the probability space $(E, F, P)$. For all $n$ , we have $$ P( X_n = 1/n ) = P(X_n = -1/n) = 1/2 $$ How can I compute the expectation of the random variable $X_n$, given $n$. I get stuck in computing the distribution function. Thanks :)

Answer 1

I don't think you can compute the distribution function. But fortunately you don't need to.

Recall that $$0.5=P(X_n=1/n)=P(A),$$ where $$A=\{\omega\in X_n:X_n(\omega)=1/n\}$$ Define $B$ similarly (for $-1/n$) so that $0.5=P(B)$. Set $E_0=E\setminus(A\cup B)$ so that $$ \mathbb{E}(X_n^+)=\int_{E_0}X_n^+(\omega)dP(\omega)+\int_AX_n^+(\omega)dP(\omega)+\int_BX_n^+(\omega)dP(\omega) $$ It should be easy to compute this as $0+0.5/n+0=0.5/n$. The computation for $\mathbb{E}(X_n^-)=0.5/n$ should be just as straightforward. Hence $$ \mathbb{E}(X_n)=\mathbb{E}(X_n^+)-\mathbb{E}(X_n^-)=0.5/n-0.5/n=0. $$